Gyrokinetic equations and their applications in magnetized plasmas

Gyrokinetic equations are powerful tools for studying collective oscillations in magnetized plasmas with realistic magnetic field geometries and plasma equilibria when the corresponding frequencies are much less than the ion cyclotron frequencies. In this tutorial lecture, we will sketch the derivation of this useful set of equations as well as provide examples of their applications in both laboratory and space plasmas. The focus of these examples will be on the spontaneous excitations (i.e., instabilities) of low-frequency magnetohydrodynamic (MHD) Alfven waves via wave-particle resonances with energetic ions.